Stop&Hop achieves early halting through an irregularity-aware Halting Policy Network \(\mathcal {H}_\theta \)  that chooses whether to Stop or Wait at a given time \(t^\prime \)  based on the history of an ongoing time series \(X\) .
If it chooses to Wait, it also predicts for how long in the form of a real-valued step size, after which it will run again.
Since there are rarely ground truth optimal halting times for when Stop&Hop should stop, we follow the state-of-the-art [1]}, [2]}, [3]} and formulate this task as a Partially-Observable Markov Decision Process (POMDP): Given a state \(h_{t^\prime }\) , select an action \(a_{t^\prime }\)  from the set {Stop, Wait}.
If \(a_{t^\prime } = \lbrace \text{Stop}\rbrace \) , then the class label for the entire series is returned at time \(t^\prime \) .
If \(a_{t^\prime } = \lbrace \text{Wait}\rbrace \) , update \(t^\prime := t^\prime + \Delta t\) , where \(\Delta t\)  is a hop-size predicted based on \(X_{\le t^\prime }\) , the observations before time \(t^\prime \) .
Overall, the earlier and more accurate the predictions are for a series \(X\) , the higher the reward.
If Stop&Hop stops too early, it may not have observed enough data and will be less likely to be accurate.
Therefore, a good solution must carefully balance between stopping and predicting the next halting time.
